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3.1: One-Step Equations

Difficulty Level: At Grade Created by: CK-12

Learning Objectives

At the end of this lesson, students will be able to:

• Solve an equation using addition.
• Solve an equation using subtraction.
• Solve an equation using multiplication.
• Solve an equation using division.

Vocabulary

Terms introduced in this lesson:

equal
equation
isolate
linear equation

Teaching Strategies and Tips

Use the introductory problem to motivate equivalent equations, since it can be solved two ways:

• The cost plus the change received is equal to the amount paid, x+22=100\begin{align*}x + 22 = 100\end{align*}.
• The cost is equal to the difference between the amount paid and the change, x=10022\begin{align*}x = 100 - 22\end{align*}

Examples 1 and 2 are essentially the same; constant terms must be added to both sides to isolate x\begin{align*}x\end{align*} on the left. Adding vertically can benefit students.

Solve 12=4+x\begin{align*}12 = -4 + x\end{align*}.

Solution. To isolate x\begin{align*}x\end{align*}, add 4\begin{align*}4\end{align*} to both sides of the equation. Add vertically.

12+416=4+x=+4=x\begin{align*}12 & = -\cancel{4} + x\\ +4 & = +\cancel{4}\\ 16 & = x\end{align*}

The variable in Example 3 is not the usual x\begin{align*}x\end{align*}. Remind students that the letter of the variable does not matter. Additional Example.

Solve 21=n+14\begin{align*}-21=n+14\end{align*}.

Hint: To isolate n\begin{align*}n\end{align*}, subtract 14\begin{align*}14\end{align*} from both sides of the equation.

In Examples 4-6, teachers may opt to solve the equations by adding the opposite in lieu of subtracting.

Example.

Solve 17=x+8\begin{align*}-17=x+8\end{align*}.

Hint: To isolate x\begin{align*}x\end{align*}, add 8\begin{align*}-8\end{align*} to both sides of the equation. Add vertically.

17825=x+8=8=x\begin{align*}-17 & = x+\cancel{8}\\ -8 & = -\cancel{8}\\ -25 & = x\end{align*}

Use Example 6 as an example of an equation with fractions. Remind students to find common denominators.

Point out in Example 8 that in general,

axb=abx\begin{align*}\frac{ax}{b}=\frac{a}{b}x\end{align*}

which will help students isolate x\begin{align*}x\end{align*} in one step, multiplying by the reciprocal of a/b\begin{align*}a/b\end{align*}.

Note that the equation in Example 10 can be written in dollars or in cents:

• 5x=3.25\begin{align*}5x=3.25\end{align*} (x\begin{align*}x\end{align*} in dollars)
• 5x=325\begin{align*}5x=325\end{align*} (x\begin{align*}x\end{align*} in cents)

Although Examples 13 and 15 can be solved by making a table and Example 14 by guessing and checking, teachers are encouraged to help students setup and solve an equation of the type presented in the lesson.

Error Troubleshooting

General Tip: After the constant term is canceled in a one-step equation, the variable must be carried down onto the next line. Remind students to write the x=\begin{align*}x =\end{align*}.

General Tip: Students forget to perform the same operation on both sides of an equation. Have students use a colored pencil to write what they are doing to both sides of the equation.

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